ok well if you did know how and you determine there was no remainder then the limit does exist
but if there is a remainder then the limit will not exist
but i guess you can also determine this from just pluggin in -4 on both top and bottom if you have 0/0 the limit will exist
if you don't then it won't

This question doesn't have a real limit had the function only exist for values where x^9+9 is greater than or equal to 0 anyhow
x^9 has to be greater to 0
which means x cannot be negative because if it makes since to say
if x is negative then x^9 is definitely more and most negative

It's not that none of the explanations don't make sense. I came to the same conclusion working it myself. It's just that when I plugged in my answer what the test told me was that, that answer is incorrect

Definitely don't rub in there face. I would just bring it up once if and when they respond.
I think an email would be just fine.
You can be like:
Hey professor,
As I was taking the exam, I came across the problem lim (x->-4) (sqrt(x^9+9)-5)/(x+4) and I said the limit does not exist but it was returned incorrect by the exam.
I was thinking it was meant to be lim (x->-4) (sqrt(x^2+9)-5)/(x+4) where the answer would be -4/5.
Please let me know if I'm correct or not.
Sincerely,
(you)
But don't say she made a mistake.
Just say the exam or whatever.
What I'm saying is don't say YOU in the email. Try not to imply it is the teacher's fault even though it is. I think that is the best way.

I think you could say just nothing exists around -4 since x^9+9 has to be greater than or equal to 0.
You know which means x^9>=-9
But for x values less than -4 certainly do not satisfy the inequality x^9>=-9.
You could actually solve that inequality if you wanted to.
x>=(-9)^(1/9)
but values really close to -4 are definitely not in the set [(-9)^(1/9),inf)